IntIntro.mws

Unit 4: Integration

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Auxiliary Plotting Commands - do not modify

>    restart;
with( plots ):
with( Student[Calculus1] ): 

Warning, the name changecoords has been redefined

>    F1 := sin(x):
a := 0:
b := Pi:
P1 := ApproximateInt( F1, x=a..b, partition=4, method=lower,
                      output=animation, refinement=halve, iterations=8,
                      title="Riemann Sums for y=sin(x) on [0,Pi]" ):

>    f1 := x^2-x+2:
P2a := plot( f1, x=0..2, color=cyan, filled=true ):
P2b := plot( f1, x=2..3, color=pink, filled=true ):
P2 := display( P2a,P2b, title="Int( f(x), x=0..3 ) = Int( f(x), x=0..2 ) + Int( f(x), x=2..3 )" ):

>    P3a := plot( x+sin(3*x), x=-Pi..0, filled=true, color=cyan ):
P3b := plot( x+sin(3*x), x= 0..Pi, filled=true, color=pink ):
P3 := display( P3a,P3b, title="Odd Function: Areas Cancel" ):

>    P4a := plot( abs(x+sin(3*x)), x=-Pi..0, filled=true, color=cyan ):
P4 := display( P4a,P3b, title="Even Function: Areas Match" ):

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>    P1;

[Maple Plot]

>    P2;

[Maple Plot]

>    P3;

[Maple Plot]

>    P4;

[Maple Plot]

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Calculus is the study of limits. Limits were introduced in Unit 1. Differentiation is the first major application of limits. In Units 2 you developed a conceptual, computational, and theoretical  understanding of derivatives. Applications of derivatives were discussed in Unit 3. Units 4 and 5 deal with another application of limits: integrals. As with derivatives, the first unit focuses on the conceptual, computational, and theoretical  understanding of integrals. Applications of the integral, including area, volume, arclength, and work, are presented in Unit 5.

There are two different types of integrals: Indefinite Integrals  and Definite Integrals . The first lessons of this unit develop both types of integral. One of the amazing mathematical results of calculus is that there are two extremely important -- and simple -- relationships between definite and indefinite integrals. These facts are stated as the (First and Second) Fundamental Theorems of Calculus.

The indefinite integral is essentially the "underivative" of a function. Pracitce applying this concept is provided by a brief introduction to Differential Equations. The definite integrals is based on adding an infinite number of numbers. The Summation Notation and Riemann Sums lesson introduces new notation that provides a good way to work with these sums. You will have the opportunity to use the Fundamental Theorems to evaluate definite integrals in the Evaluating Definite Integrals lesson.

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TO DO:

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